6277 = (2 * 3138) + 1 ;
hence not 2 divides 6277 by NAT_4:9; :: thesis:
6277 = (3 * 2092) + 1 ;
hence not 3 divides 6277 by NAT_4:9; :: thesis:
6277 = (5 * 1255) + 2 ;
hence not 5 divides 6277 by NAT_4:9; :: thesis:
6277 = (7 * 896) + 5 ;
hence not 7 divides 6277 by NAT_4:9; :: thesis:
6277 = (11 * 570) + 7 ;
hence not 11 divides 6277 by NAT_4:9; :: thesis:
6277 = (13 * 482) + 11 ;
hence not 13 divides 6277 by NAT_4:9; :: thesis:
6277 = (17 * 369) + 4 ;
hence not 17 divides 6277 by NAT_4:9; :: thesis:
6277 = (19 * 330) + 7 ;
hence not 19 divides 6277 by NAT_4:9; :: thesis:
6277 = (23 * 272) + 21 ;
hence not 23 divides 6277 by NAT_4:9; :: thesis:
6277 = (29 * 216) + 13 ;
hence not 29 divides 6277 by NAT_4:9; :: thesis:
6277 = (31 * 202) + 15 ;
hence not 31 divides 6277 by NAT_4:9; :: thesis:
6277 = (37 * 169) + 24 ;
hence not 37 divides 6277 by NAT_4:9; :: thesis:
6277 = (41 * 153) + 4 ;
hence not 41 divides 6277 by NAT_4:9; :: thesis:
6277 = (43 * 145) + 42 ;
hence not 43 divides 6277 by NAT_4:9; :: thesis:
6277 = (47 * 133) + 26 ;
hence not 47 divides 6277 by NAT_4:9; :: thesis:
6277 = (53 * 118) + 23 ;
hence not 53 divides 6277 by NAT_4:9; :: thesis:
6277 = (59 * 106) + 23 ;
hence not 59 divides 6277 by NAT_4:9; :: thesis:
6277 = (61 * 102) + 55 ;
hence not 61 divides 6277 by NAT_4:9; :: thesis:
6277 = (67 * 93) + 46 ;
hence not 67 divides 6277 by NAT_4:9; :: thesis:
6277 = (71 * 88) + 29 ;
hence not 71 divides 6277 by NAT_4:9; :: thesis:
6277 = (73 * 85) + 72 ;
hence not 73 divides 6277 by NAT_4:9; :: thesis:
6277 = (79 * 79) + 36 ;
hence not 79 divides 6277 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6277 & n is prime holds
not n divides 6277 by XPRIMET1:44;
hence 6277 is prime by NAT_4:14; :: thesis: